The Ensemble Fractal Dimension of Clouds
This page mostly fills in the mathematical derivations behind the equations in the poster, presented at AGU Fall Meeting 2022, and provides additional background info.
Clouds serve as one of the most canonical examples of fractal geometry in nature, following generalized scaling laws from the \(\sim 1\)mm scales at which turbulence convolutes air currents to planetary scales where the sphericity of the Earth prevents infinite clouds from developing. One scaling law relates the length of a cloud perimeter \(p\) to the size of the "ruler length" \(\xi\) used to measure it by introducing the fractal dimension \(D_i\) as (Mandelbrot, 1982)

\begin{align} p\propto \xi^{1-D_i}, \tag{1} \label{eq:frac D} \end{align} where \(D_i\) is the fractal dimension, \(p\) is an individual cloud's perimeter, and \(\xi\) is the size of a single pixel.

Isotropic turbulence predicts a value of \(D_i = 4/3\) for cloud perimeters (Siebesma & Jonker, 2000) with slight corrections for intermittent turbulence (Hentschel & Procaccia, 1983), which has been verified in models (Christensen & Driver, 2021; Siebesma & Jonker, 2000) and observations (Batista-Tomas et al., 2016; Lovejoy, 1982).

However, it is unknown how the total cloud perimeter \(P\), summed over all clouds in a cloud field, responds to changes in \(\xi\). The total perimeter changes because each individual clouds' perimeter changes as well as the total number of clouds.

This relationship may be characterized by an ensemble fractal dimension \(D_e\) rather than the individual fractal dimension \(D_i\) by an analogous relationship \begin{align} P\propto \xi ^{1-D_e} \tag{2} \label{eq:ens frac D} \end{align} where \(1\lt D_i\lt D_e\lt 2\) (Mandelbrot, 1982).

Furthermore, a recent study derived an expression for total cloud amount that was based on total cloud perimeter, suggesting cloud growth and decay, and ultimately areal extent, is constrainted by horizontal exchanges occurring across cloud perimeters. This expression enabled predictions of how total cloud perimeter will change under a warming climate from first-principles physical reasoning (Garrett et al., 2018). However, the link between total cloud perimeter and total cloud area, and therefore the cloud feedback, is not understood.

Here we derive an expression for the ensemble fractal dimension \(D_e\) and a relation between total cloud perimeter \(P\) and total cloud area \(A\) by assuming power law distributions for individual cloud perimeters and areas. We then compare our results to measurements from several satellite datasets.
The fractal characteristics of clouds
Numerous previous studies have shown that the distribution of cloud areas is well described by a power law distribution (Benner & Curry, 1998; Cahalan & Joseph, 1989; Christensen & Driver, 2021; Koren et al., 2008; Wood & Field, 2011; Yamaguchi & Feingold, 2013), implying a probabiliy density for cloud area \(a\) following \begin{align} n_a \equiv \frac{dn}{da} = \alpha a_\text{min}^\alpha a^{-\alpha-1}, \qquad a_\text{min}\lt a\lt a_\text{max}, \qquad 0\lt \alpha\lt 1, \tag{3} \label{eq:areadist} \end{align} where \(a_\text{max}\to\infty\) for normalization. While there is significant disagreement on the domain of validity \((a_\text{min}, a_\text{max})\), it has recently been argued much of the disagreement may be due to sampling bias and \(a_\text{min} \lt 1\)km\(^2\lt 30,000\)km\(^2 \lt a_\text{max}\), a range of scales encompasing much of the atmosphere (DeWitt, in prep). There is more agreement in measured values of \(\alpha\), which are generally between 0.8 and 0.9 (Benner & Curry, 1998; Cahalan & Joseph, 1989; Wood & Field, 2011).

Cloud perimeter distributions have only recently been measured, however. Equation \ref{eq:frac D} can be used to relate cloud area \(a\) and perimeter \(p\) (Mandelbrot, 1982) through \begin{align} p\propto \xi^{1-D_i}a^{D_i/2} \tag{4} \label{eq:p-a-relation} \end{align} assuming measured cloud areas are resolution independent. Equation \ref{eq:p-a-relation} provides a straightforward means of measuring \(D_i\) (Benner & Curry, 1998; Christensen & Driver, 2021; Lovejoy, 1982) and also implies a power law distribution of cloud perimeters together with equation \ref{eq:areadist}. Perimeters therefore follow a probability density \begin{align} n_p \equiv \frac{dn}{dp}= \beta (p_\text{min})^{\beta}p^{-\beta-1},\qquad p_\text{min}\lt p\lt p_\text{max} \qquad \beta \gt 1 \tag{5} \label{eq:pdist} \end{align} where \(\beta=2\alpha D_i^{-1}\) and again \(p_\text{max}\to \infty\) for normalization. We find \(\beta\approx 4/3\) in satellite data and \(p_\text{min}\lt 20\)km and \(p_\text{max}\gt 3000\)km when measured, respectively, with \(\xi=1\)km and \(\xi=2\)km.

An expression for total cloud perimeter as a function of measurement resolution
Consider a cloud field measured at \(\xi_1\) with \(N_\text{original}\) clouds with perimeters in the range \((p_{\text{min}_1}, \infty)\). If the pixel size is changed to \(\xi_2\) such that \(\xi_2\lt \xi_1\), there are now \(N_\text{total}\) clouds, and the increase \(N_\text{original} \to N_\text{total}\) can be thought of as combining two effects \begin{align} \frac{d\ln N}{d\ln \xi} = \left(\frac{d\ln N}{d\ln \xi}\right)_{\text{increase in domain}} + \left(\frac{d\ln N}{d\ln \xi}\right)_{\text{increase in individual perimeters}} \tag{6} \label{fndjsailbeayiul} \end{align} where the first term is due to an increase in the distribution's domain from \((p_{\text{min}_1}, \infty) \to (p_{\text{min}_2}, \infty)\), where \(p_{\text{min}_2}\lt p_{\text{min}_1}\) since at a smaller pixel size \(\xi_2\) smaller perimeters can be measured. In fact, the smallest perimeter measured must be proportional to the pixel size because the number of pixels required to accurately measure the smallest cloud is invariant under a change in resolution. The second term in equation \ref{fndjsailbeayiul} is due to an increase in the number of clouds within any given bin. Under any rescaling \(\xi_1\to\xi_2\), each individual cloud's perimeter, and therefore its location in the distribution, changes by the fractal relation. This results in a cascade of clouds moving to larger perimeter bins across the entire distribution. Because there are more clouds in bins with smaller \(p\) (equation \ref{eq:pdist}), the number of clouds within any given bin increases as \(\xi_1\to\xi_2\).

For the first term in equation \ref{fndjsailbeayiul}, consider a measurement of cloud perimeters at pixel size \(\xi_1\). When the normalized density is integrated over the range \((p_{\text{min}_1},\infty)\), it gives 1 by definition. So clearly \begin{align} N_1 = N_1\int_{p_{{\text{min}_1}}}^{\infty} n_p dp = N_1\int_{p_{{\text{min}_1}}}^{\infty} -\beta (p_{\text{min}_1})^{\beta}p^{-\beta-1} dp \tag{7} \label{eq:number} \end{align} and if the bounds of integration cover a different range in \(p\), the result will be the total number of clouds within that different range. We can therefore determine the number of clouds \(N_2\) at pixel size \(\xi_2\) by keeping the same functional form in equation \ref{eq:number} and integrating over a new domain \((p_{\text{min}_2},\infty)\) \begin{align} N_2 &= N_1\int_{p_{{\text{min}_2}}}^{\infty} -\beta (p_{\text{min}_1})^{\beta}p^{-\beta-1} dp \\ &= N_1 (p_{\text{min}_1})^{\beta} p_{\text{min}_2}^{-\beta} \end{align} or, if \(\xi_1\) is held constant, \begin{align} N_2 \propto p_{\text{min}_2}^{-\beta}. \end{align} But since \(p_{\text{min}}\propto \xi\) \begin{align} \left(\frac{d\ln N}{d\ln \xi}\right)_{\text{increase in domain}} = -\beta. \tag{8} \label{eq:dnafkebfui} \end{align}

However, equation \ref{eq:dnafkebfui} is not the whole story since every cloud's individual perimeter changes upon a rescaling of the resolution, giving the second term in equation \ref{fndjsailbeayiul}. Consider a single bin \(j\) in perimeter space within both \((p_{\text{min}_1},\infty)\) and \((p_{\text{min}_2},\infty)\) with number of clouds \(n_j\). Upon a rescaling, clouds leave \(j\) and are replaced from clouds entering from the opposite direction since the fractal relation equation \ref{eq:frac D} is monotonic. For a given resolution change, clouds move in perimeter space a distance \begin{align} \Delta \ln p = -\frac{1}{\beta} \Delta \ln n \nonumber \\ \ln p_i-\ln p_j = -\frac{1}{\beta}\ln n_i+\frac{1}{\beta}\ln n_j. \end{align} After a rescaling, however, the number of clouds in bin \(j\) is no longer \(n_j\) but instead \(n_i\) since clouds were relocated from bin \(i\) to \(j\). Thus under a rescaling \(\xi_1\to\xi_2\) the perimeter in bin \(j\) is associated with the number in bin \(i\) and \begin{align} p\propto n^{\frac{1}{\beta}} \end{align} for any perimeter bin. Using the fractal relation, \begin{align} n\propto \xi^{\beta -\beta D_i} \end{align} and \begin{align} \frac{d\ln n}{d\ln \xi}=\left(\frac{d\ln N}{d\ln \xi}\right)_{\text{increase in individual perimeters}} = \beta-\beta D_i, \end{align} giving in total for equation \ref{fndjsailbeayiul} \begin{align} \frac{d\ln N}{d\ln \xi} = -\beta D_i. \tag{9} \label{nfeukabh} \end{align}

The total perimeter \(P\) is obtained by integrating the weighted perimeter density \begin{align} P &= N\int_{p_\text{min}}^\infty pn_p dp \\ &= N\beta p_\text{min}^{\beta} \int_{p_{\text{min}}}^\infty p^{-\beta} dp \tag{10} \label{eq:normalized total P} \\ &= N\frac{\beta}{\beta-1} p_\text{min}. \tag{11} \label{eq:totalP} \end{align}

Since \(p_\text{min}\propto\xi\), equation \ref{nfeukabh} implies \(N\propto \xi^{-\beta D_i}\) \begin{align} P\propto \xi^{-\beta D_i+1}. \end{align}
Linking total cloud perimeter to total cloud area
Equation \ref{eq:totalP} implies \(P\propto N\), assuming \(p_\mathrm{min}\) is approximately the Kolmogorov microscale as \(\xi\to 0\) and is therefore unchanged in a warmer climate. Since \(\alpha\) in equation \ref{eq:areadist} is less than 1, the total cloud area diverges as \(a_\text{max}\to\infty\) and therefore largest cloud determines the total area. \begin{align} A &= N \int_{a_\text{min}}^{a_\text{max}}an_ada \\ &= N \int_{a_\text{min}}^{a_\text{max}}\alpha a_\text{min}^\alpha a^{-\alpha}da \\ &= N \frac{\alpha}{1-\alpha} a_\text{min}^\alpha a^{-\alpha+1}\big|_{a_\text{min}}^{a_\text{max}} \\ &= N \frac{\alpha}{1-\alpha} a_\text{min}^\alpha \left(a_\text{max}^{-\alpha+1} -a_\text{min}^{-\alpha+1} \right). \end{align} Using equation \ref{eq:totalP}, \begin{align} A = \frac{P}{p_\text{min}} \frac{\beta-1}{\beta}\frac{\alpha}{1-\alpha} a_\text{min}^\alpha \left(a_\text{max}^{-\alpha+1} -a_\text{min}^{-\alpha+1} \right). \end{align} The exponent \(-\alpha+1\) is \(1/9\) if \(\beta=D_i=4/3\), which makes the terms \(a_\text{max}^{-\alpha+1}\) and \(a_\text{min}^{-\alpha+1}\) comparable in magnitude unless the range between \(a_\text{max}\) and \(a_\text{min}\) is extremely large. The range of scales over which a satellite can resolve clouds, about 4-5 orders of magnitude, is not large enough, but the true range of atmospheric cloud areas likely is so we can neglect \(a_\text{min}\) if we consider the entire atmospheric range and \begin{align} A&\approx P \frac{\alpha(\beta-1)}{\beta(1-\alpha)} \frac{a_\mathrm{min}^\alpha}{p_\text{min}} a_\mathrm{max}^{-\alpha+1}. \end{align}

Assuming \(a_\mathrm{min}\) remains constant under a warming climate as we expect \(p_\text{min}\) to, then approximately \(A\propto N \propto P\) due to the weak dependence on \(a_\mathrm{max}\). Therefore, unless the area of the largest cloud changes dramatically, if \(P\) increases under a warming climate, so too should total cloud area under the following assumptions:

Both cloud perimeters and areas follow a power law distribution such that over a range of scales large enough that the upper bound in equations \ref{eq:pdist} and \ref{eq:areadist} can be approximated as \(\infty\).

The perimeter distribution (equation \ref{eq:pdist}) satisfies \(\beta\gt 1\) and the area distribution (equation \ref{eq:areadist}) \(0\lt \alpha\lt 1\).

Clouds with perimeters or areas outside of the power law range in equations \ref{eq:pdist} and \ref{eq:areadist} contribute negligible perimeter/area (\(p\) or \(a\)) to the total (\(P\) or \(A\)).

Clouds within the power law distribution have, on average, a constant fractal dimension \(D_i\). This is implied by the first assumption but deserves to be made explicit.

The effects of clouds combining or separating as the pixel size \(\xi\) changes are negligible or implicit in the calculation of the individual fractal dimension \(D_i\).